In recent years, the Norlund sum and its applications have attracted considerable attention, especially in mathematics and other applied sciences. Because this sum plays an important role in mathematical modeling involving differential equations and difference equations, as well as in the theory of infinite series and integral transforms. Therefore, one of the most important goals of this article is to present a new approach to generating function and to derive new formulas including the sum of (inverse) Laplace transforms, the Norlund sum, the alternative Hurwitz zeta function, and the inverse Catalan sum formula with related operators. Another aim is to establish the relationship between the inverse Catalan sum formula and the derivative operator, and to give new formulas and relations, including the zeta function, the Bernoulli numbers, and the Laplace transform. Furthermore, applications are given including the relationship between these operators and the Dirichlet L-function. Some new formulas for certain classes of infinite series, including the Dirichlet L-function, the zeta function, Euler numbers of the second kind, and the Apostol-Bernoulli polynomials of higher-order.

In this paper, by virtue of the beta polynomials, we introduce and study a new special word family referred to as the beta-based words. In order to construct the beta-based words and their generation algorithm, we inspire from the definition of the beta polynomials. By making implementation in the Wolfram language, we obtain some tables for these words and their decimal equivalents. We also show that the decimal equivalents of the beta-based words are associated with the Mersenne numbers. For their visual illustrations, we also present plots designed by black-white superimposed square blocks emerging from the first few beta-based words. Moreover, we investigate some properties of the beta-based words, involving length, height, slope, symmetry and complement properties. Additionally, in this paper, we construct a generalization of harmonic numbers. We also derive some combinatorial sums related to the beta polynomials, the Bernstein polynomials and the generalized harmonic numbers.

In this paper, we develop a higher-order theory for the numbers \( y_{9,n}^{(\alpha)}(\lambda;a) \) and the associated polynomials \( y_{9,n}^{(\alpha)}(x,\lambda;a) \), extending several identities and interpolation formulas previously obtained for the case \( \alpha=1 \). Using the corresponding generating functions, we derive explicit representations, recurrence relations, and structural identities, and we establish connections with a broad range of classical and modern special numbers, including the Apostol--Bernoulli, Apostol--Euler, Euler--Frobenius, Fubini, and Stirling numbers. We further construct an interpolation function for the higher-order numbers and prove that its special values at negative integers reproduce these numbers up to an explicit normalization factor. A residue-class decomposition of this interpolation function is then obtained, leading naturally to a generalized hypergeometric Hurwitz--Lerch-type zeta function. Finally, we study several special cases and analytic properties of this zeta-type function, including its reduction to the classical Lerch transcendent and a differential identity.

We consider the N-tuple Hurwitz-Lerch zeta function introduced by Srivastava and Choi, ζN(s,x,λ)=∑k1,…,kN≥0λk1+⋯+kN(x+k1+⋯+kN)s, where N is a positive integer and λ is a complex parameter. A central breakthrough in this work is the explicit reduction of the N-tuple function to a linear combination of single Lerch transcendents via a finite-difference identity, which naturally leads to sharp multiplication and inversion formulas. We further investigate the combinatorial structure of the underlying polynomial coefficients, establishing their symmetry, unimodality, log-concavity, and precise asymptotic behavior. As key applications, we derive new identities for higher-order Apostol-Euler and Apostol-Bernoulli polynomials. Our results unify and extend several known formulas.

In this paper, we investigate fractional B splines and their connections with Fourier analysis, and establish connections with generalized Stirling-type numbers and distribution theory. Employing a generating function approach inspired by recent results of Simsek [24], we derive a novel Fourier type expansion for fractional B splines that involves generalized Stirling type numbers. Our main contribution is the derivation of a Fourier-type expansion of fractional B splines in terms of generalized Stirling-type numbers. This representation allows us to express fractional B splines as infinite linear combinations of derivatives of the Dirac delta in the distributional sense. Furthermore, we establish an explicit shifted distributional representation and obtain shifted distributional representations that characterize the action of fractional B-splines on test functions. In addition, we introduce a new class of fractional spline polynomials and derive their generating function in terms of the Mittag Leffler function. These results provide a unified framework that connects spline theory, fractional calculus, and combinatorial structures.

The first goal of this paper is to give a new family of special function related to the Debye functions. By using these function and generating function method, we derive many novel formulas and relations involving the Debye functions, multiple Hurwitz Lerch zeta function which interpolate the Apostol-Bernoulli numbers of higher order at negative integers. Proof method of theorems in this paper are different from those of theory of analytic numbers. The second goal of this paper is to show that special values and integral presentations of this function are closely related to unification of the Debye functions, the moment generating function for the negative binom distribution, the generating functions for the Apostol-type numbers of higher-order, and the Frobenius-Euler numbers, the Bernoulli numbers, and the Stirling numbers etc. Moreover, we give recurrence relations, Maclaurin's series expansion, and approximation formula for unification of the Debye functions. Finally, we give further remarks, observations and applications in mathematical physics on the results of this paper.

This paper is structured around three main objectives. First, to derive some new relations between Bernstein basis functions and the Legendre basis functions by means of generating functions, including derivative operators and Rodrigues formula. Second, to give relations between Bernstein basis functions and Bernoulli polynomials by applying the Nörlund sum to the generating function for Bernstein basis functions. These relations involving certain infinite series in terms of the Bernstein basis functions and Bernoulli polynomials and partial derivative equations. Third, to give formulas for multivariable Bernstein basis functions, the Touchard polynomials, the Bernoulli numbers, the Stirling numbers, etc., by using functional equations, generating functions, and the Euler derivative operator. Some of the results of this paper are analyzed with comments on how to reduce them to known results.

The aim of this paper is to apply the Norlund sum to Bessel function and generalized Bernoulli polynomials involving this function, we derive some new relations and integral formulas. By applying the Laplace transform with its inverse to generating function for the generalized Bernoulli polynomials involving the Bessel function, we also derive infinite series representation for these polynomials. Furthermore, some relationship between the Bessel function and the Norlund sum is found. We also give some new relations by comparing specific values to the Bessel function and modified Bessel functions with the aid of their Laplace transforms by its inverse.

In this paper, with inspiration of the definition of Bernstein basis functions and their recurrence relation, we give construction of a new word family that we refer Bernstein-based words. By classifying these special words as the first and second kinds, we investigate their some fundamental properties involving periodicity and symmetricity. Providing schematic algorithms based on tree diagrams, we also illustrate the construction of the Bernstein-based words. For their symbolic computation, we also give computational implementations of Bernstein-based words in the Wolfram Language. By executing these implementations, we present some tables of Bernstein-based words and their decimal equivalents. In addition, we present black-white and four-colored patterns arising from the Bernstein-based words with their potential applications in computational science and engineering. We also give some finite sums and generating functions for the lengths of the Bernstein-based words. We show that these functions are of relationships with the Catalan numbers, the centered m$$ m $$-gonal numbers, the Laguerre polynomials, certain finite sums, and hypergeometric functions. We also raise some open questions and provide some comments on our results. Finally, we investigate relationships between the slopes of the Bernstein-based words and the Farey fractions.

The purpose of this paper is to modify generating functions for the Frobenius–Euler type Simsek numbers and polynomials. We also give some alternating generating functions for these numbers and polynomials. By using these modification generating functions, we derive many novel functional equations involving not only the Frobenius–Euler type Simsek numbers and polynomials, but also other certain classes of special numbers and polynomials. By applying binomial series with multinomial binomial coefficients to these equations, we also derive many novel and useful formulae. By the aid of moment generating functions for the geometric distribution, we give novel relations and formulae among the Frobenius–Euler type Simsek numbers and polynomials, the Apostol–Bernoulli numbers, the Stirling numbers, the moments of the geometric distribution, and other special numbers and polynomials. We also give inequalities involving the Bernoulli numbers and polynomials, the Ostrowski inequality, and the generalized Euler type identity.

We study various types of generating functions for higher-order derangement numbers. We present a generating function for a new class of binomial-type polynomials with coefficients given by higher-order derangement numbers. Using this generating function, we derive explicit formulas for these polynomials, as well as derivative and integral formulas involving Stirling numbers and Cauchy numbers of the first kind. By employing this generating function along with other special formulas, we obtain several new results involving higher-order derangement numbers, Bernoulli numbers, Stirling numbers, combinatorial numbers associated with Daehee numbers, Peters-type Simsek numbers, and special finite sums.

The first aim of this paper is to present fundamental properties of the Nörlund sum (or the Nörlund sum operator) with their applications. The second purpose of this paper is to find not only some new formulas involving some certain classes of polynomials and finite sums with aid of the Nörlund sum, but also novel multiplication formula for the Nörlund sum. Moreover, by applying this multiplication formula to analytic functions and meromorphic functions, we derive the Bernoulli polynomials and a multiplication formula for the Hurwitz zeta function. We show that special case of this multiplication formula gives the Raabe’s multiplication formula for the Bernoulli polynomials. Finally, we give relations among the Nörlund sum, indefinite sum, and the Volkenborn integral.

The goal of this paper is to define a new approach when constructing generating functions for the generalization and unification of the Apostol type Bernoulli polynomials. We apply the N & ouml;rlund sum, the Euler operator for derivative, ad the (inverse) Laplace transform to reach this aim. We also aim to prove a functional equation involving the N & ouml;rlund sum and the (inverse) Laplace transform. Moreover, by combining these operators with functional equations of the generating functions, we derive some new formulas for the Apostol type polynomials and kth moment of the geometric distribution. Finally, applying these operators, we not only find new the Riemann integral formulas, but also construct interpolation functions related to the Lerch zeta and the Hurwitz zeta functions for these polynomials. (c) 2025 Elsevier Inc. All rights are reserved, including those for text and data

One of the main motivations of this paper is to construct generating functions for generalization of the Touchard polynomials (or generalization exponential functions) and certain special numbers. Many novel formulas and relations for these polynomials are found by using the Euler derivative operator and functional equations of these functions. Some novel relations among these polynomials, beta polynomials, Bernstein polynomials, related to Binomial distribution from discrete probability distribution classes, are given.

The goal of this manuscript is to present a new sequence in connection with finite sum involving harmonic numbers. We give some novel formulas containing some known certain special finite sums which are (inverse) binomial coefficients, the harmonic numbers, special numbers and polynomials.

. We investigate and study on mathematical structures involving mathematical models and others associated with seismic waves in an earthquake. Our first aim is to give some novel formulas and certain finite sums including the Bernoulli numbers and the Hermite polynomials with the aid of generating functions, the Riemann integral, and the Volkenborn integral. The second aim is to examine the seismic wave propagation in different geological units with the help of special polynomials containing the Hermite polynomials and their graph fitting of functions. To evaluate the shape of the seismic waves propagating within the ground (rock and/or soil), we use comparing method with the graph of the Hermite polynomials and functions and the polynomial Rocking Bearings. Furthermore, we also define generating function for the polynomial type Rocking Bearings. We give open problems on this generating function and earthquake facts. By applying partial derivative operator to the generating function for the m-parametric Hermite type polynomials, we give a novel recurrence relation and derivative formulas for these polynomials. We also give a new general formula for monomials in terms of these polynomials. Moreover, for the purpose of visualizing curve fitting approach to the seismic waves, we draw many plots of the Hermite functions with Mathematica (Version 12.0.0) with their codes. Finally, with the aid of these graphs, we give useful evaluation on the shapes of the seismic waves propagated in the ground (rock and/or soil).

By using the Barnes' type multiple partial Euler zeta function, which is interpolate the (h, q)-extension of Euler polynomials at negative integers, we give some identities associated with (h, q)-extension of Euler polynomials and numbers and the Apostol-Bernoulli numbers of higher order.

Not only are there operators for studying properties for special numbers and polynomials, but the Volkenborn integral has an equally powerful applications. The aim of this article is to derive new formulas by applying operators and ($p$-adic)the Volkenborn integral to certain families polynomial, especially the Euler polynomials. These formulas include the Stirling numbers, array polynomials, the Fubini type polynomials, and the Bernoulli and Euler numbers and polynomials.

The aim of this paper is to derive many novel formulas involving the sum of powers of consecutive integers, the Bernoulli polynomials, the Stirling numbers and moments arise from conditional probability, moment generating functions and arithmetic functions by using the methods and techniques, which are used in discrete distributions in statistics such as uniform distribution, moment generating functions, and other probability distributions. Moreover, relations among the generalized Euler totient function, finite distributions containing special numbers and polynomials, discrete probability formula, and other special functions are given. By using the Riemann zeta function and the Liouville function, we derive a novel moment formula for probability distribution on the set positive integers. Finally, by using approximation formulas for certain family of finite sums, we derive formulas not only for the sum of powers of consecutive integers involving the Bernoulli polynomials, but also for the conditional probability involving the Laplace rule of succession.